We now consider the XY model with a quantum quench, assuming the sudden quench approximation.
Here, following the original work of [1]}, we consider an XY model environment
coupled to a two-level central spin-1/2 system. Due to this coupling, the wave function of the
XY model evolves in two distinct branches, as the central spin backreacts differently on the
environment depending on whether it is in the ground state \(\mathinner {|{g}\rangle }\)  or the excited state \(\mathinner {|{e}\rangle }\) .
We will take the interaction Hamiltonian to be \(H_I\) , so that the total Hamiltonian is \(H_F=H+H_I\) ,
with \(H\)  given in Eq. (REF ), and
\(H_I &=& c_1\frac{\delta }{2}\mathinner {|{e}\rangle }\mathinner {\langle {e}|}\sum \limits _{l=1}^{N}\sigma ^{z}_{l} \nonumber \\&-& c_2\frac{\delta }{4}\mathinner {|{e}\rangle }\mathinner {\langle {e}|}\sum \limits _{l=1}^{N}\left(\sigma ^{x}_{l}\sigma ^{x}_{l+1}-\sigma ^{y}_{l}\sigma ^{y}_{l+1}\right)~.\) 
